Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
366 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #8805 | C2 | 1 | 6 | yes | maximal |
| 2 | #8131 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #1842 | C22 | 1 | 4 | yes | — |
| 4 | C42×C8 | C22 | 1 | 3 | yes | — |
| 5 | C2×#334 | — | 2 | 12 | no | — |
| 6 | #1273 | C4 | 1 | 4 | yes | — |
| 7 | C2×C4⋊C8 | — | 2 | 12 | no | — |
| 8 | C2×C4×C8 | C2×C4 | 1 | 6 | yes | — |
| 9 | C2×C4×C8 | D4 | 1 | 6 | yes | — |
| 10 | C2×C4×C8 | Q8 | 1 | 6 | yes | — |
| 11 | C43 | C23 | 1 | 1 | yes | characteristic |
| 12 | #334 | — | 4 | 24 | no | — |
| 13 | C22×C8 | — | 4 | 4 | no | — |
| 14 | C22×C8 | C4⋊C4 | 1 | 12 | yes | — |
| 15 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 16 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 17 | C2×C42 | C2×Q8 | 1 | 3 | yes | — |
| 18 | C4⋊C8 | — | 4 | 24 | no | — |
| 19 | C4×C8 | — | 2 | 12 | no | — |
| 20 | C4×C8 | — | 4 | 6 | no | — |
| 21 | C4×C8 | SD16 | 1 | 24 | yes | — |
| 22 | C22×C4 | C2×C4⋊C4 | 1 | 3 | yes | — |
| 23 | C22×C4 | #127 | 1 | 1 | yes | characteristic |
| 24 | C22×C4 | C4⋊Q8 | 1 | 3 | yes | — |
| 25 | C2×C8 | — | 2 | 12 | no | — |
| 26 | C2×C8 | — | 2 | 12 | no | — |
| 27 | C2×C8 | — | 8 | 12 | no | — |
| 28 | C2×C8 | #106 | 1 | 24 | yes | — |
| 29 | C42 | — | 2 | 6 | no | — |
| 30 | C42 | #131 | 1 | 4 | yes | — |
| 31 | C42 | C2×SD16 | 1 | 12 | yes | — |
| 32 | C23 | #384 | 1 | 1 | yes | characteristic |
| 33 | C2×C4 | — | 2 | 6 | no | — |
| 34 | C2×C4 | #424 | 1 | 6 | yes | — |
| 35 | C2×C4 | C2×#106 | 1 | 6 | yes | — |
| 36 | C2×C4 | #430 | 1 | 6 | yes | — |
| 37 | C2×C4 | #492 | 1 | 6 | yes | — |
| 38 | C2×C4 | #498 | 1 | 6 | yes | — |
| 39 | C8 | — | 2 | 24 | no | — |
| 40 | C8 | — | 16 | 4 | no | — |
| 41 | C22 | #1843 | 1 | 1 | yes | characteristic |
| 42 | C22 | #1844 | 1 | 3 | yes | — |
| 43 | C22 | #1849 | 1 | 3 | yes | — |
| 44 | C4 | — | 2 | 6 | no | — |
| 45 | C4 | #1570 | 1 | 12 | yes | — |
| 46 | C4 | #1705 | 1 | 4 | yes | — |
| 47 | C2 | #8809 | 1 | 3 | yes | — |
| 48 | C2 | #8813 | 1 | 1 | yes | characteristic |
| 49 | C2 | #8825 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| socle, prime radical | C23 | #384 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #8805 | 2 |
| #8131 | 2 |